Theorem sb4bor 1833

Description: Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)

Assertion

Ref	Expression
sb4bor	|- ( A. x x = y \/ A. x ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )

Proof of Theorem sb4bor

Step	Hyp	Ref	Expression
1		sb4or 1831	. 2 |- ( A. x x = y \/ A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
2		sb2 1765	. . . . 5 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3		df-bi 117	. . . . . 6 |- ( ( ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) -> ( ( [ y / x ] ph -> A. x ( x = y -> ph ) ) /\ ( A. x ( x = y -> ph ) -> [ y / x ] ph ) ) ) /\ ( ( ( [ y / x ] ph -> A. x ( x = y -> ph ) ) /\ ( A. x ( x = y -> ph ) -> [ y / x ] ph ) ) -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) ) )
4	3	simpri 113	. . . . 5 |- ( ( ( [ y / x ] ph -> A. x ( x = y -> ph ) ) /\ ( A. x ( x = y -> ph ) -> [ y / x ] ph ) ) -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )
5	2, 4	mpan2 425	. . . 4 |- ( ( [ y / x ] ph -> A. x ( x = y -> ph ) ) -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )
6	5	alimi 1453	. . 3 |- ( A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) -> A. x ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )
7	6	orim2i 761	. 2 |- ( ( A. x x = y \/ A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) ) -> ( A. x x = y \/ A. x ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) ) )
8	1, 7	ax-mp 5	1 |- ( A. x x = y \/ A. x ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   A.wal 1351   [wsb 1760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761

This theorem is referenced by: (None)